I think the same would happen with:
What is the hat you prefer?
Well, now that I am writing it out, I think this would be understood...
It would probably be understood, but it sounds like foreign English, and you are right that this is a good example of how the Spanish sounds.
Some people (of other dialects and/or education levels) may say "What hat do you prefer'," but it is not standard English, and is considered incorrect in formal writing. But no one, to my knowledge, would say it as you wrote it above.
As a mnemonic, we might memorize this, said by a police officer:
¿Cuál es tu nombre, y qué es eso que tienes en la mano'
The first question asks for a fact to be stated, while the second question asks for a description. Note that both are translated as "what."